This invention relates to the field of finite element simulations, specifically finite element simulations of problem spaces comprising two regions meshed with non-conforming meshes.
Finite element analysis often requires connection of two meshes at a shared boundary. For example, finite element analysis of a system comprising separate subsystem models can require connection of the subsystem model meshes. One approach to connecting the meshes requires that both meshes have the same number of nodes, the same nodal coordinates, and the same interpolation functions at the shared boundary. If these requirements are met, then the two meshes can be connected simply by equating the degrees of freedom of corresponding nodes at the shared boundary. Ensuring conformity between meshes in this manner can require a significant amount of time and effort in mesh generation.
A simple alternative to such an approach is to use node collocation, also known as tied contact or standard multi-point constraints, to connect the meshes. With this approach, one of the connecting mesh surfaces is designated as the master surface and the other as the slave surface. For problems in solid mechanics, the meshes are connected by constraining nodes on the slave surface to specific points on the master surface at all times. Although this approach is appealing because of its simplicity, overlaps and gaps may develop between the two meshes either because of non-planar initial geometry or non-uniform displacements. For example, a node on the master surface may either penetrate or pull away from the slave surface during deformation even though the slave node constraints are all satisfied. As a result, displacement continuity may hold only at specific points on the master-slave interface.
Dissimilar meshes can also be connected using two-field or three-field hybrid methods. See, e.g., Farhat and Geradin, xe2x80x9cUsing a Reduced Number of Lagrange Multipliers for Assembling Parallel Incomplete Field Finite Element Approximationsxe2x80x9d, Computer Methods in Applied Mechanics and Engineering 97, 333-354, (1992); Aminpour et al., xe2x80x9cA Coupled Analysis Method for Structures with Independently Modelled Finite Element Subdomainsxe2x80x9d, International Journal for Numerical Methods in Engineering 38, 3695-3718 (1995). For two-field methods, the master and slave surface displacements and Lagrange multipliers for constraints between these displacements are considered as field-variables. For three-field methods, the displacement of an interface surface is introduced as an additional field-variable. Discretizations of Lagrange multipliers are required by both methods. For three-field methods, a discretization of the interface displacement is also required. The use of Lagrange multipliers leads to indefinite systems of equations for both methods. As a result, sparse solvers based on the Cholesky decomposition cannot be applied directly to such systems.
Another approach to connecting dissimilar meshes involves defining new shape functions for elements on a slave surface to ensure displacement continuity with the master surface. A specialized application of this approach was used to develop a quadrilateral transition element for mesh refinement. See, e.g., Hughes, The Finite Element Method, Prentice-Hall, Englewood Cliffs, N.J. (1987). A systematic method to generate the new shape functions can be developed based on the concepts of vertex modes, edge modes, and face modes used by the p version of finite elements. See, e.g., Szabo and Babuska, Finite Element Analysis, John Wiley and Sons, New York, N.Y. (1991). Although the new shape functions and their spatial derivatives can be determined in a straightforward manner, it can be very difficult to devise simple and accurate numerical integration rules for the volume integrals defining element matrices. This difficulty is especially evident when two or more faces of an element are connected to a master surface. The major hindrance to the development of the integration rules is the piecewise definition of the new shape functions.
The present invention provides a method of connecting dissimilar finite element meshes. A first mesh, designated the master mesh, and a second mesh, designated the slave mesh, each have interface surfaces proximal the other. Each interface surface has a corresponding interface mesh comprising a plurality of interface nodes. Each slave interface node is assigned new coordinates locating the interface node on the interface surface of the master mesh. The slave interface surface is further redefined to be the projection of the slave interface mesh onto the master interface surface. The present invention provides the freedom to designate master and slave surfaces independently of the resolutions of the two meshes. Standard practice is to designate the master surface as the surface with fewer numbers of nodes. Improved accuracy can be achieved in certain instances by allowing the master surface to have the greater number of nodes. Methods of mesh refinement based on adaptive subdivision of existing elements can also benefit from this flexibility. For example, kinematic constraints on improper nodes can be removed while preserving the convergence characteristics of adjacent elements.
Advantages and novel features will become apparent to those skilled in the art upon examination of the following description or may be learned by practice of the invention. The objects and advantages of the invention may be realized and attained by means of the instrumentalities and combinations particularly pointed out in the appended claims.